We are continuing to invent, develop, and translate novel Magnetic Resonance (MR) based water displacement imaging methods. Diffusion Tensor MRI (DT-MRI or DTI) is perhaps the best-known imaging method that we invented and developed to date. It measures a diffusion tensor of mobile water within tissue. It consists of relating an effective diffusion tensor to the measured MR spin echo signal; estimating an effective diffusion tensor, D, in each pixel from a set of diffusion-weighted MR images; and calculating and displaying information derived from D. This information includes the local fiber-tract orientation, the mean-squared displacement water molecules migrate or diffuse in any given direction, the orientationally-averaged mean diffusivity, and other scalar invariant quantities that are independent of the laboratory coordinate system. These scalar parameters are intrinsic properties of the tissue and we measure them without contrast agents or dyes. For example, one DTI-derived quantity, the orientationally-averaged diffusivity (or Trace), has been the most successful imaging parameter used to date to visualize an acute stroke in progress. Subsequently, showed that DTI is effective in identifying Wallerian degeneration often associated with chronic stroke. Previous studies with kittens showed DTI to be useful in following early developmental changes occurring in cortical gray and white matter, which are not detectable using other means, and which became the basis for applying these approaches in humans. The development of a method to color-encode nerve fiber orientation in the brain by Sinisa Pajevic and Carlo Pierpaoli has allowed us to identify and differentiate anatomical white matter pathways that have similar structure and composition, but different spatial orientations. These Direction-Encoded Color (DEC) maps of the human brain clearly show the main association, projection, and commissural white matter pathways, and are a mainstay in modern Neuroradiology practice and can be seen in publications like Gray's Anatomy. To assess anatomical connectivity between different functional regions in the brain, we also proposed and demonstrated a way to use DTI data to trace out nerve fiber tract trajectories, which we called DTI tractography. This was made possible by the development and implementation of a general mathematical framework for obtaining a continuous, smooth approximation to the measured discrete, noisy, diffusion tensor field data by Sinisa Pajevic and Akram Aldroubi. Collectively, these methods and approaches have allowed us and many other groups world-wide to perform detailed anatomical and structural analyses of the brain in vivo, which was only possible previously using laborious, invasive histological methods performed on dead tissue. As we migrated DTI to large, multi-center and multi-patient studies, we began developing a variety of statistical techniques to interpret our imaging findings quantitatively, specifically to be able to determine the statistical significance of differences observed in our DTI data. To this end, we developed empirical Monte Carlo and Bootstrap methods for determining features of the statistical distribution of the diffusion tensor from experimental DTI data. Another innovation was a novel tensor-variate Gaussian distribution that describes the variability of the diffusion tensor in an idealized DTI experiment, and can be used to optimize the design and efficiency of DTI experiments. More recently, we developed approaches to measure uncertainties of many tensor-derived quantities, including the direction of nerve pathways using perturbation and statistical approaches. These collective developments provide the foundation for the use of powerful hypothesis tests to address a wide variety of important biological and clinical questions that previously could only be tackled using ad hoc methods, if at all. More recently, we have been developing sophisticated mathematical/physical models of water diffusion profiles and related these to the MR signals that we measure, with the aim of using our MRI data to infer new microstructural features of tissue (primarily white matter in the brain). One example of this idea was our composite hindered and restricted model of diffusion (CHARMED) MRI framework. A more recent enhancement of CHARMED, AxCaliber MRI, is another. AxCaliber allows us to measure the axon diameter distribution within a nerve bundle and the volume fraction of axons within that bundle from MR displacement imaging data. Sophisticated multiple pulsed field gradient (PFG) NMR and MRI sequences, developed by Michael Komlosh, help us characterize microscopic anisotropy within tissues like gray matter that are macroscopically isotropic, appearing like a homogeneous and featureless gel in DTI. She and Ferenc Horkay have developed physical phantoms to test and interrogate our mathematical models of water diffusion in complex tissues developed by Evren Ozarslan. Evren also developed novel ways to interpret data obtained from the MR sequences to learn more about the size, shape, and distribution of pores in biological tissue and other porous media. He has also used advanced mathematical techniques to characterize anomalous diffusion observed in various tissue specimen that are indicative of an underlying fractal architecture. Parameters derived from these novel measurements may provide a new source of MR contrast for promising neuroscience applications, such as in vivo (Brodmann or cytoarchitechtonic) parcellation of the cerebral cortex or clinical diagnostic applications, such as improved cancer detection and tumor staging. He has also developed novel approaches to characterize non-Gaussian features of the displacement distribution measured using MRI. To this end, our group continues to work on reconstructing the average propagator (displacement distribution) or features of it, using a relatively small number of diffusion weighted images (DWI). The average propagator is the holy grail of displacement or diffusion imaging, which can by used to infer geometric features of microscopic restricted compartments as well as glean all of the information provided by DTI as well as other higher-order tensor (HOT) methods. One approach we used previously was to use an iterative reconstruction scheme along with a priori information and physical constraints to infer the average propagator from DWI data. Another approach was to use CT-like reconstruction methods to estimate the displacement profile from DWI data. The most successful method to date, however, developed by Evren Ozarslan, uses a Hermite function expansion of the average propagator. This dramatically reduces the amount of DWI data required while providing a plethora of new imaging parameters or stains with which to characterize microstructural features in tissue. Collectively, these novel methods and methodologies represent a pathway to realizing in vivo MRI histology--providing detailed microstructural and microarchitectural information about cells and tissues that otherwise could only be obtained using laborious and invasive histological or pathological techniques applied on biopsied or excised specimens. We continue to develop new ways to assess tissue structure and architecture in vivo and non-invasively, with the aim of translating these approaches to the clinic, and to the larger biomedical research community, which we have done successfully with DTI.